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Physical Review A

Strong damping and low-temperature anomalies for the harmonic oscillator

Fritz Haake, Reinhard Reibold

Published Oct 1, 1985• 410 citations
Publisher

Abstract

We consider Ullersma's model of a damped harmonic oscillator interacting with a heat bath. We show that for large damping the oscillator does not undergo Brownian motion at any temperature. For high temperatures non-Markovian effects show up in initial slips only. For low temperatures we find, as further anomalies, power-law decay of time-dependent expectation values as well as a strong squeezing of the displacement and momentum uncertainties; again these anomalies are most strongly pronounced for strong damping.

DOI
10.1103/physreva.32.2462

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We consider Ullersma's model of a damped harmonic oscillator interacting with a heat bath. We show that for large damping the oscillator does not undergo Brownian motion at any temperature. For high temperatures non-Markovian effects show up in initial slips only. For low temperatures we find, as further anomalies, power-law decay of time-dependent expectation values as well as a strong squeezing of the displacement and momentum uncertainties; again these anomalies are most strongly pronounced for strong damping.
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The analysis of a strongly coupled harmonic quantum Brownian motion has been performed in [1] for a special class of spectral densities obtained as a generalization of the Drude model. In the present scenario, we extend the study of the strongly coupled harmonic quantum Brownian motion to regular spectral densities that are structured as sub-Ohmic at low frequencies and arbitrarily shaped at high frequencies. The bosonic environment is initially in the vacuum state unentangled from the coherent state of the main oscillator. As a generalization of the previous results, we obtain that the long time dynamics is determined uniquely by the initial condition and the low frequency structure of the spectral density. Also in the present framework, inverse power law regressions to the asymptotics appear. The position and the momentum tend to undamped oscillations. The number of excitations relaxes to its initial value with damped oscillations enveloped in inverse power law relaxations. For the momentum and the number of excitations the inverse power law decays become arbitrarily slow in critical configurations by approaching the upper bound of the sub-Ohmic regime. The critical frequencies of the main oscillator are determined by the first negative moment of the spectral densities. Differences with respect to the weak coupling regime arise.
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The exact dynamics of a quantum damped harmonic oscillator coupled to a reservoir of boson modes has been formally described in terms of the coupling function, both in weak and strong coupling regime. In this scenario, we provide a further description of the exact dynamics through integral transforms. We focus on a special class of spectral densities, sub-ohmic at low frequencies, and including integrable divergencies referred to as photonic band gaps. The Drude form of the spectral densities is recovered as upper limit. Starting from special distributions of coherent states as external reservoir, the exact time evolution, described through Fox H-functions, shows long time inverse power law decays, departing from the exponential-like relaxations obtained for the Drude model. Different from the weak coupling regime, in the sub-ohmic condition, undamped oscillations plus inverse power law relaxations appear in the long time evolution of the observables position and momentum. Under the same condition, the number of excitations shows trapping of the population of the excited levels and oscillations enveloped in inverse power law relaxations. Similarly to the weak coupling regime, critical configurations give arbitrarily slow relaxations useful for the control of the dynamics. If compared to the value obtained in weak coupling condition, for strong couplings the critical frequency is enhanced by a factor 4.
We investigate the exact quantum dynamics of a free particle damped through its interaction with an harmonic bath, when the effective coupling strength behaves as ωδ at low frequency. We find that various regimes can occur depending on the value of δ and of the temperature. At large times, the mean square displacement is shown either to diverge as tv, the exponent v being never greater than 2 or to tend towards a finite value, indicating a confinement in the broad sense. In addition, for δ < 1, an oscillation (absent in the uncoupled system) is found : due to the friction on the dominant low-frequency modes and to the existence of retardation effects, the particle is forced to relax in the mean towards its initial location, the net dynamical effect of the bath being similar to that of an applied potential. In the limit δ → 0, the particle is frozen.
Time evolution of a harmonic oscillator linearly coupled to a heat bath is compared for three classes of initial states for the bath modes-grand canonical ensemble, number states, and coherent states. It is shown that for a wide class of number states the behavior of the oscillator is similar to the case of the equilibrium bath. If the bath modes are initially in coherent states, then the variances of the oscillator coordinate and momentum, as well as its entanglement to the bath, asymptotically approach the same values as for the oscillator at zero temperature and the average coordinate and momentum show a Brownian-like behavior. We derive an exact master equation for the characteristic function of the oscillator valid for arbitrary factorized initial conditions. In the case of the equilibrium bath this equation reduces to an equation of the Hu-Paz-Zhang type, while for the coherent states bath it leads to an exact stochastic master equation with a multiplicative noise.